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Using the Fundamental Theorem of Line
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- A. State the Fundamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardEvaluate the line integral ∫CF→⋅dr→ using the Fundamental Theorem of Line Integrals if F→(x,y)=(4x+4y)i→+(4x+4y)j→and Cis the smooth curve from (−1,1)to (5,6). Enter the exact answer. ∫CF→⋅dr→=arrow_forwardUsing Stokes' theorem, solve the line integral of G(x, y, z) - (1, x + yz, xy-√z) around the boundary of surface S, which is given by the piece of the plane 3x + 2y + z = 1 where x, y, and z all ≥ 0.arrow_forward
- Use Stokes' Theorem to calculate the line integral of the vector function F = (y, z, x) along the curve C, which is obtained by the intersection of the surfaces x + y = 2 and x² + y² + z² = 2 ( x + y).arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 2)arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4arrow_forward
- Use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwise indicated. ?C(lnx+y)dx−x2dy where C is the rectangle with vertices (1, 1), (3, 1), (1, 4), and (3, 4)arrow_forwardEvaluate the line integrals using the Fundamental Theorem of Line Integrals: ∫c (yi+xj)*dr Where C is any path from (0,0) to (2,4).arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forward
- Evaluate C F · dr using the Fundamental Theorem of Line Integrals. F(x, y, z) = 2xyzi + x2zj + x2yk C: smooth curve from (0, 0, 0) to (1, 3, 2)arrow_forwardEvaluate C F · dr using the Fundamental Theorem of Line Integrals. F(x, y) = e2xi + e2yj C: line segment from (−1, −1) to (0, 0)arrow_forwardThe figure shows a vector field F and two curves C_1 and C_2. Are the line integrals of F over C_1 and C_2 positive, negative, or zero? Explain.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning